/[MITgcm]/manual/s_algorithm/text/tracer.tex
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revision 1.18 by edhill, Sat Oct 16 03:40:13 2004 UTC revision 1.24 by cnh, Thu Jan 17 17:42:11 2008 UTC
# Line 359  if the limiter is set to zero. Line 359  if the limiter is set to zero.
359    
360    
361  \section{Non-linear advection schemes}  \section{Non-linear advection schemes}
362    \label{sect:non-linear_advection_schemes}
363  \begin{rawhtml}  \begin{rawhtml}
364  <!-- CMIREDIR:non-linear_advection_schemes: -->  <!-- CMIREDIR:non-linear_advection_schemes: -->
365  \end{rawhtml}  \end{rawhtml}
# Line 627  as if in one dimension: Line 628  as if in one dimension:
628  \tau^{n+1/3} & = & \tau^{n}  \tau^{n+1/3} & = & \tau^{n}
629  - \Delta t \left( \frac{1}{\Delta x} \delta_i F^x(\tau^{n})  - \Delta t \left( \frac{1}{\Delta x} \delta_i F^x(\tau^{n})
630             + \tau^{n} \frac{1}{\Delta x} \delta_i u \right) \\             + \tau^{n} \frac{1}{\Delta x} \delta_i u \right) \\
631  \tau^{n+2/3} & = & \tau^{n}  \tau^{n+2/3} & = & \tau^{n+1/3}
632  - \Delta t \left( \frac{1}{\Delta y} \delta_j F^y(\tau^{n+1/3})  - \Delta t \left( \frac{1}{\Delta y} \delta_j F^y(\tau^{n+1/3})
633             + \tau^{n} \frac{1}{\Delta y} \delta_i v \right) \\             + \tau^{n} \frac{1}{\Delta y} \delta_i v \right) \\
634  \tau^{n+3/3} & = & \tau^{n}  \tau^{n+3/3} & = & \tau^{n+2/3}
635  - \Delta t \left( \frac{1}{\Delta r} \delta_k F^x(\tau^{n+2/3})  - \Delta t \left( \frac{1}{\Delta r} \delta_k F^x(\tau^{n+2/3})
636             + \tau^{n} \frac{1}{\Delta r} \delta_i w \right)             + \tau^{n} \frac{1}{\Delta r} \delta_i w \right)
637  \end{eqnarray}  \end{eqnarray}
# Line 664  $W$: {\bf rTrans} (local) Line 665  $W$: {\bf rTrans} (local)
665    
666  \end{minipage} }  \end{minipage} }
667    
668    \begin{figure}
669    \resizebox{3.5in}{!}{\includegraphics{part2/multiDim_CS.eps}}
670    \caption{Muti-dimensional advection time-stepping with Cubed-Sphere topology
671    \label{fig:advect-multidim_cs}
672    }
673    \end{figure}
674    
675  \section{Comparison of advection schemes}  \section{Comparison of advection schemes}
676    \label{sect:tracer_advection_schemes}
677  \begin{rawhtml}  \begin{rawhtml}
678  <!-- CMIREDIR:comparison_of_advection_schemes: -->  <!-- CMIREDIR:comparison_of_advection_schemes: -->
679  \end{rawhtml}  \end{rawhtml}
# Line 677  $W$: {\bf rTrans} (local) Line 685  $W$: {\bf rTrans} (local)
685     Advection Scheme & code & use  & use Multi- & Stencil & comments \\     Advection Scheme & code & use  & use Multi- & Stencil & comments \\
686                      &      & A.B. & dimension & (1 dim) & \\                      &      & A.B. & dimension & (1 dim) & \\
687     \hline \hline     \hline \hline
688       $1^{rst}$order upwind  & 1 &  No & Yes & 3 pts & linear/$\tau$, non-linear/v\\
689       \hline
690     centered $2^{nd}$order & 2 &  Yes & No & 3 pts & linear \\     centered $2^{nd}$order & 2 &  Yes & No & 3 pts & linear \\
691     \hline     \hline
692     $3^{rd}$order upwind   & 3 &  Yes & No & 5 pts & linear/tracer\\     $3^{rd}$order upwind   & 3 &  Yes & No & 5 pts & linear/$\tau$\\
693     \hline     \hline
694     centered $4^{th}$order & 4 &  Yes & No & 5 pts & linear \\     centered $4^{th}$order & 4 &  Yes & No & 5 pts & linear \\
695     \hline \hline     \hline \hline
696  %  Lax-Wendroff       & 10 &  No & Yes & 3 pts & linear/tracer, non-linear/flow\\     $2^{nd}$order DST (Lax-Wendroff)  & 20 &
697  %  \hline                           No & Yes & 3 pts & linear/$\tau$, non-linear/v\\
698     $3^{rd}$order DST & 30 &  No & Yes & 5 pts & linear/tracer, non-linear/flow\\     \hline
699       $3^{rd}$order DST & 30 &  No & Yes & 5 pts & linear/$\tau$, non-linear/v\\
700     \hline \hline     \hline \hline
701     $2^{nd}$order Flux Limiters & 77 &  No & Yes & 5 pts & non-linear \\     $2^{nd}$order Flux Limiters & 77 &  No & Yes & 5 pts & non-linear \\
702     \hline     \hline
# Line 695  $W$: {\bf rTrans} (local) Line 706  $W$: {\bf rTrans} (local)
706   \caption{Summary of the different advection schemes available in MITgcm.   \caption{Summary of the different advection schemes available in MITgcm.
707            ``A.B.'' stands for Adams-Bashforth and ``DST'' for direct space time.            ``A.B.'' stands for Adams-Bashforth and ``DST'' for direct space time.
708            The code corresponds to the number used to select the corresponding            The code corresponds to the number used to select the corresponding
709            advection scheme in the parameter file (e.g., {\em tempAdvScheme=3} in            advection scheme in the parameter file (e.g., {\bf tempAdvScheme}=3 in
710            file {\em data} selects the $3^{rd}$ order upwind advection scheme            file {\em data} selects the $3^{rd}$ order upwind advection scheme
711            for temperature).            for temperature).
712     }     }
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